International J.Math. Combin. Vol.1(2018), 97-108
Minimum Equitable Dominating Randić Energy of a Graph Rajendra P. (Bharathi College, PG and Research Centre, Bharathinagara, 571 422, India)
R.Rangarajan (DOS in Mathematics, University of Mysore, Mysuru, 570 006, India)
prajumaths@gmail.com, rajra63@gmail.com
Abstract: Let G be a graph with vertex set V (G), edge set E(G) and di is the degree of its i-th vertex vi , then the Randić matrix R(G) of G is the square matrix of order n, whose (i, j)-entry is equal to √ 1 if the i-th vertex vi and j-th vertex vj of G are adjacent, di dj
and zero otherwise. The Randic energy [3] RE(G) of the graph G is defined as the sum of the absolute values of the eigenvalues of the Randić matrix R(G). A subset ED of V (G) is called an equitable dominating set [11], if for every vi ∈ V (G) − ED there exists a vertex
vj ∈ ED such that vi vj ∈ E(G) and |di (vi ) − dj (vj )| ≤ 1. In the contrast, such a dominating
set ED is Smarandachely if |di (vi ) − dj (vj )| ≥ 1. Recently, Adiga, et.al. introduced, the
minimum covering energy Ec(G) of a graph [1] and S. Burcu Bozkurt, et.al. introduced, Randić Matrix and Randić Energy of a graph [3]. Motivated by these papers, Minimum equitable dominating Randić energy of a graph REED (G) of some graphs are worked out and bounds on REED (G) are obtained.
Key Words: Randić matrix and its energy, minimum equitable dominating set and minimum equitable dominating Randić energy of a graph.
AMS(2010): 05C50. §1. Introduction Let G be a graph with vertex set V (G) and edge set E(G). The adjacency matrix A(G) of the graph G is a square matrix, whose (i, j)-entry is equal to 1 if the vertices vi and vj are adjacent, otherwise zero [12]. Since A(G) is symmetric, its eigenvalues are all real. Denote them by λ1 , λ2 , . . . , λn , and as a whole, they are called the spectrum of G and denoted by Spec(G). The energy of graph [12] G is ε(G) =
Pn
i=1
|λi |.
The literature on energy of a graph and its bounds can refer [4,8,9,10,12]. The Randić matrix R(G) = (rij ) of G is the square matrix of order n, where 1 Received
July 28, 2017, Accepted March 1, 2018.
98
Rajendra P. and R.Rangarajan
(rij ) =
√1
di dj
0,
, if vi and vj are adjacent vertices in G; otherwise.
The Randic energy [3] RE(G) of the graph G is defined as the sum of the absolute values of the eigenvalues of the Randić matrix R(G). Let ρ1 , ρ2 , · · · , ρn be the eigenvalues of the Randić matrix R(G). Since R(G) is symmetric matrix, these eigenvalues are real numbers and their sum is zero. Randić energy [3] can be defined as Pn RE(G) = i=1 |ρi |
For details of Randić energy and its bounds, can refer [2, 3, 5, 6, 7]. Let G be a graph with vertex set V (G) = {v1 , v2 , · · · , vn } and ED is minimum equitable dominating set of G. Minimum equitable dominating Randić matrix of G is n × n matrix RED (G) = (rij ), where 1 √di dj , if vi and vj are adjacent vertices in G; (rij ) = 1, if i = j and vi ∈ ED; 0, otherwise. The characteristic polynomial of RED (G) is denoted by det(ρI−RED (G)) = |ρI−RED (G)|. Since RED (G) is symmetric, its eigenvalues are real numbers. If the distinct eigenvalues of RED (G) are ρ1 > ρ2 > · · · > ρr with their multiplicities are m1 , m2 , · · · , mr then spectrum of RED (G) is denoted by ρ1 ρ2 · · · ρr . SpecRED (G) = m1 m2 · · · mr The minimum equitable dominating Randić energy of G is defined as REED (G) =
n X i=1
|ρi |.
Example 1.1 Let W5 be a wheel graph, with vertex set V (G) = {v1 , v2 , v3 , v4 , v5 }, and let its minimum equitable dominating set be ED = {v1 }. Then minimum equitable dominating Randić matrix RED (W5 ) is
1
√1 12 1 RED (W5 ) = √12 1 √ 12 √1 12
√1 12
√1 12
0
√1 12 1 3
1 3
0
1 3
0
1 3
0
1 3
0
1 3
0
√1 12 1 3
0 1 3 0
−0.6666 0 0.2324 1.4342 . SpecRED (W5 ) = 1 2 1 1
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Minimum Equitable Dominating Randić Energy of a Graph
The minimum equitable dominating Randić energy of W5 is REED (W5 ) = 2.3332.
§2. Bounds for the Minimum Equitable Dominating Randić Energy of a Graph Lemma 2.1 If ρ1 , ρ2 , · · · , ρn are the eigenvalues of RED (G). Then n X i=1
and
n X i=1
ρi = |ED|
ρ2i = |ED| + 2
X 1 , dd i<j i j
where ED is minimum equitable dominating set. Proof (i) The sum of eigenvalues of RED (G) is n X
ρi =
i=1
n X i=1
rii = |ED|.
(ii) Consider, the sum of squares of ρ1 , ρ2 , . . . , ρn is, n X
ρ2i
=
n X n X i=1 j=1
i=1
=
n X i=1
n X i=1
rij rji =
ρ2i
n X
(rii )2 +
i=1
(rii )2 + 2
X
rij rji
i6=j
X (rij )2 i<j
X 1 = |ED| + 2 . dd i<j i j
2
Upper and lower bounds for REED (G) is similar proof to McClelland’s inequalities [10], are given below Theorem 2.2(Upper Bound) Let G be a graph with ED is minimum equitable dominating set. Then v u u X 1 u REED (G) ≤ tn |ED| + 2 . d dj i i<j Proof Let ρ1 , ρ2 , · · · , ρn be the eigenvalues of RED (G). By Cauchy-Schwartz inequality, we have !2 ! n ! n n X X X b2i , (1) ai b i ≤ a2i i=1
where a and b are any real numbers.
i=1
i=1
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Rajendra P. and R.Rangarajan
If ai = 1, bi = |ρi | in (1), we get n X i=1
!2
|ρi |
≤
n X
12
i=1
by Lemma 2.1,
!
n X
|ρi |2
i=1
X 1 , [REED (G)]2 ≤ n |ED| + 2 d d i j i<j
!
v u u X 1 u REED (G) ≤ tn |ED| + 2 . d d i<j i j
Theorem 2.3(Lower Bound) Let G be a graph with |ED| is minimum equitable dominating set and di is degree of vi . Then REED (G) ≥ where D =
Qn
i=1
s
|ED| + 2
X 1 2 + n(n − 1)D n , dd i<j i j
|ρi |.
Proof Consider 2
[REED (G)] =
"
n X i=1
#2
|ρi |
=
n X i=1
|ρi |2 +
X i6=j
|ρi | |ρj |.
(2)
By using arithmetic and geometric mean inequality, we have 1 n(n − 1)
X i6=j
X i6=j
X i6=j
|ρi | |ρj | |ρi | |ρj | |ρi | |ρj |
≥ ≥ ≥
1 n(n−1) Y |ρi | |ρj |
i6=j
n(n − 1) n(n − 1)
n Y
|ρi |
n Y
|ρi |
i=1
i=1
2(n−1)
!2/n
1 ! n(n−1)
.
Now using (3) in (2), we get 2
[REED (G)] ≥
n X i=1
2
|ρi | + n(n − 1)
[REED (G)]2 ≥ |ED| + 2 REED (G) ≥
s
|ED| + 2
n Y
i=1
!2/n
|ρi |
,
n Y X 1 2 + n(n − 1)D n , where D = |ρi |, dd i<j i j i=1
X 1 2 + n(n − 1)D n . d d i j i<j
(3)
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Minimum Equitable Dominating Randić Energy of a Graph
§3. Bounds for Largest Eigenvalue of RED (G) and its Energy
Proposition 3.1 Let G be a graph and ρ1 (G) = max1≤i≤n {|ρi |} be the largest eigenvalue of RED (G). Then v u s u X 1 X 1 u1 t |ED| + 2 ≤ ρ1 (G) ≤ |ED| + 2 . n dd dd i<j i j i<j i j Proof Consider, ρ21 (G) ρ1 (G)
=
max1≤i≤n {|ρi |2 } ≤
≤
s
|ED| + 2
X 1 dd i<j i j
n X i=1
|ρi |2 = |ED| + 2
X 1 dd i<j i j
Next, n ρ21 (G) ≥ ρ21 (G) ≥
ρ1 (G) ≥ Therefore,
n X
X 1 dd i=1 i<j i j X 1 1 |ED| + 2 n d d i j i<j v u u X 1 u1 t |ED| + 2 n d d i j i<j ρ2i = |ED| + 2
v u s u X X 1 1 u1 t |ED| + 2 ≤ ρ1 (G) ≤ |ED| + 2 . n dd dd i<j i j i<j i j
Proposition 3.2 If G is a graph and n ≤ |ED| + 2 REED (G) ≤
|ED| + 2
P
1 i<j di dj
n
P
1 i<j di dj ,
v u u X u + t(n − 1) |ED| + 2 i<j
then
1 − di dj
|ED| + 2
P
n
1 i<j di dj
!2 .
Proof We know that, n X i=1
ρ2i = |ED| + 2
X 1 , dd i<j i j
n X i=2
ρ2i = |ED| + 2
X 1 − ρ21 d d i j i<j
(4)
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Rajendra P. and R.Rangarajan
By Cauchy-Schwarz inequality, we have n X
ai b i
i=2
!2
≤
n X
a2i
i=2
!
n X
b2i
i=2
!
.
If ai = 1 and bi = |ρi |, we have n X i=2
!2
|ρi |
≤
2
≤
REED (G)
≤
[REED (G) − |ρ1 |]
(n − 1)
n X i=2
|ρi |2
X 1 (n − 1) |ED| + 2 − ρ21 d d i j i<j v u u X 1 u ρ1 + t(n − 1) |ED| + 2 − ρ21 d d i j i<j
Consider the function, v u u X 1 u F (x) = x + t(n − 1) |ED| + 2 − x2 . d d i j i<j Then,
p x (n − 1) F (x) = 1 − q . P |ED| + 2 i<j di1dj − x2 ′
Here F (x) is decreasing in
X X 1 1 1 |ED| + 2 , |ED| + 2 . n d d d dj i j i i<j i<j
We know that F ′ (x) ≤ 0,
We have
p x (n − 1) 1− q ≤ 0. P |ED| + 2 i<j di1dj − x2 x≥
Since,
s
|ED| + 2
P
n
1 i<j di dj
.
P X 1 |ED| + 2 i<j n ≤ |ED| + 2 and dd n i<j i j
1 di dj
≤ ρ1 ,
(5)
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Minimum Equitable Dominating Randić Energy of a Graph
we have s
|ED| + 2
P
1 i<j di dj
n
≤
|ED| + 2
P
1 i<j di dj
n
≤ ρ1 ≤ |ED| + 2
X 1 . dd i<j i j
Then, equation (5) become
REED (G) ≤
|ED| + 2
P
n
1 i<j di dj
v u u X 1 u + t(n − 1) |ED| + 2 − dd i<j i j
|ED| + 2
P
1 i<j di dj
n
!2 .
2
§4. Minimum Equitable Dominating Randić Energy of Some Graphs Theorem 4.1 If Kn is complete graph with n vertices, then minimum equitable dominating Randić energy of Kn is 3n − 5 REED (Kn ) = . n−1 Proof Let Kn be the complete graph with vertex set V (G) = {v1 , v2 , . . . , vn } and minimum equitable dominating set is ED = {v1 }, we have characteristic polynomial of RED (Kn ) is
ρ − 1
−1
n−1
−1
n−1
. |ρI − RED (Kn )| =
..
−1
n−1
−1
n−1
−1
n−1
−1 n−1
−1 n−1 −1 n−1
···
.. .
ρ .. .
··· .. .
−1 n−1 −1 n−1 −1 n−1
−1 n−1 −1 n−1 −1 n−1
···
ρ −1 n−1
···
··· ···
−1 n−1 −1 n−1 −1 n−1
.. .
−1 n−1 −1 n−1 −1 n−1
.. .
ρ
−1 n−1
−1 n−1 −1 n−1
−1 n−1
ρ
−1
n−1
−1
n−1
−1
n−1
..
.
−1
n−1
−1
n−1
ρ
.
n×n
1 Rk′ = Rk − R2 , k = 3, 4, · · · , n − 1, n. Then, we get ρ + n−1 common from R3 to Rn and we have
ρ − 1 −1 −1 −1 −1 −1
· · ·
n−1 n−1 n−1 n−1 n−1
−1 −1 −1 −1
−1 ρ · · · n−1 n−1 n−1
n−1
n−1
0
−1 1 · · · 0 0 0
n−2
.
1 . . . . . .
. . . . . . . . |ρI − RED (Kn )| = ρ + . . . . . .
. n−1
0 −1 0 ··· 1 0 0
0 −1 0 ··· 0 1 0
0 −1 0 ··· 0 0 1
n×n
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Rajendra P. and R.Rangarajan
C2′ = C2 + C3 + · · · + Cn , we get the characteristic polynomial |ρI − RED (Kn )| = SpecRED (Kn ) =
n−2 1 2n − 3 n−3 2 ρ+ ρ − ρ+ , n−1 n−1 n−1 √ √
−1 n−1
(2n−3)− 4n−3 2(n−1)
n−2
1
(2n−3)+ 4n−3 2(n−1)
1
.
The minimum equitable dominating Randić energy of Kn is REED (Kn ) =
3n − 5 . n−1
2
Theorem 4.2 If Sn , (n ≥ 4) is star graph with n vertices, then minimum equitable dominating Randić energy of Sn is REED (Sn ) = n. Proof Let Sn , (n ≥ 4) be the star graph with vertex set V (G) = {v1 , v2 , · · · , vn } and minimum equitable dominating set is ED = {v1 , v2 , · · · , vn }, we have characteristic polynomial of RED (Sn ) is
ρ − 1
√−1
n−1
√−1
n−1
. |ρI − RED (Sn )| = ..
−1
√
n−1
−1
√n−1
−1
√n−1
√−1 n−1
√−1 n−1
···
√−1 n−1
√−1 n−1
ρ−1
0
···
0
0
0 .. .
ρ−1 .. .
··· .. .
0 .. .
0 .. .
0
0
···
ρ−1
0
0
0
···
0
ρ−1
0
0
···
0
0
√−1
n−1
0
0
..
.
0
0
ρ − 1
.
n×n
Rk′ = Rk − R2 , k = 3, 4, · · · , n. Then taking (ρ − 1) common from R3 to Rn , we get
ρ − 1
√−1
n−1
0
. n−2
|ρI − RED (Sn )| = (ρ − 1)
..
0
0
0
√−1 n−1
√−1 n−1
···
√−1 n−1
√−1 n−1
ρ−1
0
···
0
0
−1 .. .
1 .. .
··· .. .
0 .. .
0 .. .
−1
0
···
1
0
−1
0
···
0
1
0
0
−1
0
···
C2′ = C2 + C3 + · · · + Cn . Then, the characteristic polynomial |ρI − RED (Sn )| = ρ(ρ − 1)n−2 (ρ − 2),
√−1
n−1
0
0
..
.
0
0
1
.
n×n
105
Minimum Equitable Dominating Randić Energy of a Graph
SpecRED (Sn ) =
0
2 . 1
1
1 n−2
2
The minimum equitable dominating Randić energy of Sn is REED (Sn ) = n.
Theorem 4.3 If Km,n , where m < n and |m − n| ≥ 2 is complete bipartite graph with m + n vertices, then minimum equitable dominating Randić energy of Km,n is REED (Km,n ) = m + n. Proof Let Km,n , where m < n and |m − n| ≥ 2 be the complete bipartite graph with vertex set V (G) = {v1 , v2 , . . . , vm , u1 , u2 , · · · , un } and minimum equitable dominating set is ED = V (G), we have characteristic polynomial of RED (Km,n ) is
0
ρ − 1
ρ−1
0
..
..
. .
0 0
0 0 |ρI − RED (Km,n )| =
−1 −1
√mn √mn
√−1
mn √−1 mn
.. ..
. .
−1
√ √−1 mn
mn
−1
√mn √−1 mn
√−1 mn −1 √ mn
√−1 mn −1 √ mn
···
√−1 mn √−1 mn
√−1 mn √−1 mn
···
√−1 mn √−1 mn
ρ−1
0
···
0 .. .
ρ−1 .. .
··· .. .
√−1 mn √−1 mn
0
0
···
0
0
···
...
0
0
... .. .
0 .. .
0 .. .
··· ρ− 1 ···
0 ρ−1
0 √−1 mn √−1 mn
··· ··· .. .
.. .
.. .
√−1 mn √−1 mn
··· ···
.. .
.. .
√−1 mn −1 √ mn
··· .. .
.. .
√−1
mn
√−1
mn
..
.
−1 −1 √ √
mn mn
√−1 √−1
mn mn
. 0 0
0 0
.. ..
. .
ρ−1 0
0 ρ − 1
...
Rk′ = Rk − Rm , k = 1, 2, 3, · · · , m − 1 and Rd′ = Rd − Rm+1 , d = m + 2, m + 3, · · · , m + n. Then, taking (ρ − 1) common from R1 to Rm−1 and Rm+2 to Rm+n , we get
1
0
..
.
0
m+n−2
0 (ρ − 1)
√−1
mn
0
..
.
0
0
0
···
0
−1
1 .. .
··· .. .
0 .. .
−1 .. .
0
···
1
−1
0 √−1 mn
···
0
0
0 .. .
0 .. .
··· .. . ···
0
0
√−1 mn
√−1 mn
√−1 mn
ρ−1
ρ−1
···
0
0 0 .. . 0
···
√−1 mn
0
···
0
···
√−1 mn
0 .. .
··· .. .
0 .. .
0 .. .
−1 .. .
1 .. .
··· .. .
0 .. .
0
···
0
0
−1
0
···
1
0
0
0
···
−1
0
···
0
0
0
..
.
0
√−1
mn
. 0
0
..
.
0
1
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Rajendra P. and R.Rangarajan
′ Cm+1 = Cm+1 + Cm+2 + · · · + Cm+n ,
1
0
..
.
0
m+n−2
0 (ρ − 1)
√−1
mn
0
..
.
0
0
0
···
0
−1
0
0
···
0
1 .. .
··· .. .
0 .. .
−1 .. .
0 .. .
0 .. .
··· .. .
0 .. .
0
···
1
−1
···
···
0
0
0
0
√−n mn
√−1 mn
√−1 mn
···
√−1 mn
ρ−1
0
···
0
ρ−1
0
√−1 mn
···
√−1 mn
0 .. .
··· .. .
0 .. .
0 .. .
0 .. .
1 .. .
··· .. .
0 .. .
0
···
0
0
0
0
···
1
0
···
0
0
0
0
···
0
The characteristic polynomial |ρI − RED (Km,n )| = ρ (ρ − 1)m+n−2 (ρ − 2), 0 SpecRED (Km,n ) = 1
1
2
m+n−2 1
0
0
..
.
0
√−1
mn
. 0
0
..
.
0
1
.
The minimum equitable dominating Randić energy of Km,n is REED (Km,n ) = m + n.
2
Theorem 4.4 If Kn×2 , (n ≥ 3) is cocktail party graph with 2n vertices, then minimum equitable dominating Randić energy of Kn×2 is 4n−6 n−1 .
Proof Let Kn×2 , (n ≥ 3) be the cocktail party graph with vertex set V (G) = {v1 , v2 , · · · , vn , u1 , u2 , · · · , un } and minimum equitable dominating set is ED = {v1 , u1 }, we have characteristic polynomial of RED (Kn×2 ) is
λ − 1
0
−1
2n−2
−1
2n−2
.
.
. |ρI − RED (Kn×2 )| =
−1
2n−2
−1
2n−2
.
..
−1
2n−2
−1
2n−2
0
−1 2n−2
−1 2n−2
λ−1
−1 2n−2
−1 2n−2
−1 2n−2
λ
−1 2n−2
−1 2n−2
−1 2n−2
.. .
.. .
λ .. .
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
.. .
0 .. .
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
.. .
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
−1 2n−2
0
··· .. .
−1 2n−2
−1 2n−2
.. .
.. .
···
λ
··· .. .
−1 2n−2
··· ··· ···
··· ··· ···
−1 2n−2 −1 2n−2 −1 2n−2
··· .. .
−1 2n−2
−1 2n−2
···
−1 2n−2
.. .
λ .. .
··· .. .
···
−1 2n−2
−1 2n−2
···
···
0
−1 2n−2
···
.. . −1 2n−2
.. . λ −1 2n−2
−1
2n−2
−1
2n−2
−1
2n−2
−1
2n−2
..
.
.
0
−1
2n−2
..
.
−1
2n−2
λ
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Minimum Equitable Dominating Randić Energy of a Graph
′ Rk′ = Rk − R3 , k = 4, 5, · · · , n and Rn+k = Rn+k − Rk+1 , k = 2, 3, · · · , n, we get
λ − 1
0
−1
2n−2
0
.
.
. |ρI−RED (Kn×2 )| =
0
0
.
..
0
0
0
−1 2n−2
−1 2n−2
λ−1
−1 2n−2
−1 2n−2
λ
−1 2n−2
−1 2n−2
0 .. .
−1 2n−2
0
−1 2n−2
.. .
−λ
λ+
−1 2n−2
−1 2n−2
−1 2n−2
···
−1 2n−2
0
··· .. .
0 .. .
−1 2n−2
···
1 2n−2
.. .
−λ
−1 2n−2
···
0
···
−1 2n−2
··· ···
−1 2n−2
··· .. .
0 .. .
−1 2n−2
···
0
.. .
1 2n−2
λ+
−1 2n−2
···
0 .. .
−λ .. .
0 .. .
··· .. .
0 .. .
λ .. .
··· .. .
0 .. .
0
0
0
···
0
0
···
λ
0
0
0
···
−λ
0
···
0
−1
2n−2
−1
2n−2
−1
2n−2
. 1
2n−2
0
..
.
0
λ
0 .. .
C3′ = C3 + C4 + · · · + Cn + Cn+1 + · · · + C2n and Ck′ = Ck + Cn+(k−1) , k = 4, 5, · · · , n + 1. We get
λ − 1
0
−1
2n−2
0
.
.
. |ρI−RED (Kn×2 )| =
0
0
.
..
0
0
0
−1
λ−1
−1
−1 2n−2
λ−
−1 n−1 −1 n−1
(2n−4) (2n−2)
−1 2n−2
−1 n−1
−1 2n−2
···
−1 n−1
0
··· .. .
0 .. .
−1 2n−2
···
−1 n−1 1 n−1
λ+
−1 n−1
···
−1 2n−2
···
−1 2n−2
··· ···
−1 2n−2
··· .. .
0 .. .
−1 2n−2
···
0
0 .. .
0 .. .
0
0
0
···
0 .. .
0 .. .
0 .. .
··· .. .
0 .. .
λ .. .
··· .. .
0 .. .
0
0
0
···
0
0
···
λ
0
0
0
···
0
0
···
0
.. .
λ+
.. .
1 n−1
|ρI − RED (Kn×2 )| = SpecRED (Kn×2 ) =
ρ
(ρ − 1) ρ +
−1 n−1
0
n−2 n−1
1 n−1
n−2
2
ρ −
1
√ (2n−3)− 4n−3 2(n−1)
1
1
2n − 3 n−1
n−3 ρ+ n−1 √
(2n−3)+ 4n−3 2(n−1)
1
−1
2n−2
−1
2n−2
. 1
2n−2
0
..
.
0
λ
0 .. .
The characteristic polynomial n−1
−1
2n−2
,
.
The minimum equitable dominating Randić energy of Kn×2 is REED (Kn×2 ) = where n ≥ 3.
4n − 6 , n−1
2
108
Rajendra P. and R.Rangarajan
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